Format:
Online-Ressource (XII, 405p. 2 illus, digital)
ISBN:
9783642150074
Series Statement:
Probability and Its Applications
Content:
Preface -- 1.Introduction -- 2.Fundamentals of Stein's Method -- 3.Berry-Esseen Bounds for Independent Random Variables -- 4.L^1 Bounds -- 5.L^1 by Bounded Couplings -- 6 L^1: Applications -- 7.Non-uniform Bounds for Independent Random Variables -- 8.Uniform and Non-uniform Bounds under Local Dependence -- 9.Uniform and Non-Uniform Bounds for Non-linear Statistics -- 10.Moderate Deviations -- 11.Multivariate Normal Approximation -- 12.Discretized normal approximation -- 13.Non-normal Approximation -- 14.Extensions -- References -- Author Index -- Subject Index -- Notation.
Content:
Since its introduction in 1972, Stein’s method has offered a completely novel way of evaluating the quality of normal approximations. Through its characterizing equation approach, it is able to provide approximation error bounds in a wide variety of situations, even in the presence of complicated dependence. Use of the method thus opens the door to the analysis of random phenomena arising in areas including statistics, physics, and molecular biology. Though Stein's method for normal approximation is now mature, the literature has so far lacked a complete self contained treatment. This volume contains thorough coverage of the method’s fundamentals, includes a large number of recent developments in both theory and applications, and will help accelerate the appreciation, understanding, and use of Stein's method by providing the reader with the tools needed to apply it in new situations. It addresses researchers as well as graduate students in Probability, Statistics and Combinatorics.
Note:
Includes bibliographical references and indexes
,
""Preface""; ""Contents""; ""Introduction""; ""The Central Limit Theorem""; ""A Brief History of Stein's Method""; ""The Basic Idea of Stein's Method""; ""Outline and Summary""; ""Fundamentals of Stein's Method""; ""Stein's Equation""; ""Properties of the Solutions""; ""Construction of Stein Identities""; ""Sums of Independent Random Variables""; ""Exchangeable Pairs""; ""Zero Bias""; ""Size Bias""; ""A General Framework for Stein Identities and Normal Approximation for Lipschitz Functions""; ""Appendix""; ""Berry-Esseen Bounds for Independent Random Variables""
,
""Normal Approximation with Lipschitz Functions""""The Lindeberg Central Limit Theorem""; ""Berry-Esseen Inequality: The Bounded Case""; ""The Berry-Esseen Inequality for Unbounded Variables""; ""The Concentration Inequality Approach""; ""An Inductive Approach""; ""A Lower Berry-Esseen Bound""; ""L1 Bounds""; ""Sums of Independent Variables""; ""L1 Berry-Esseen Bounds""; ""Contraction Principle""; ""Hierarchical Structures""; ""Bounds to the Normal for Approximately Linear Recursions""; ""Normal Bounds for Hierarchical Sequences""; ""Convergence Rates for the Diamond Lattice""
,
""Cone Measure Projections""""Coupling Constructions for Coordinate Symmetric Variables and Their Projections""; ""Construction and Bounds for Cone Measure""; ""Combinatorial Central Limit Theorems""; ""Use of the Exchangeable Pair""; ""Construction and Bounds for the Combinatorial Central Limit Theorem""; ""Simple Random Sampling""; ""Chatterjee's L1 Theorem""; ""Locally Dependent Random Variables""; ""Smooth Function Bounds""; ""Fast Rates for Smooth Functions""; ""Appendix""; ""Linfty by Bounded Couplings""; ""Bounded Zero Bias Couplings""; ""Exchangeable Pairs, Kolmogorov Distance""
,
""Size Biasing, Kolmogorov Bounds""""Size Biasing and Smoothing Inequalities""; ""Linfty: Applications""; ""Combinatorial Central Limit Theorem""; ""Uniform Distribution on the Symmetric Group""; ""Distribution Constant on Conjugacy Classes""; ""Doubly Indexed Permutation Statistics""; ""Patterns in Graphs and Permutations""; ""The Lightbulb Process""; ""Anti-voter Model""; ""Binary Expansion of a Random Integer""; ""Discretized Normal Approximation""; ""Poisson Binomial""; ""Sum of Independent Integer Valued Random Variables""; ""Non-uniform Bounds for Independent Random Variables""
,
""A Non-uniform Concentration Inequality""""Non-uniform Berry-Esseen Bounds""; ""Uniform and Non-uniform Bounds Under Local Dependence""; ""Uniform and Non-uniform Berry-Esseen Bounds""; ""Outline of Proofs""; ""Applications""; ""Uniform and Non-uniform Bounds for Non-linear Statistics""; ""Introduction and Main Results""; ""Applications""; ""U-statistics""; ""Multi-sample U-statistics""; ""L-statistics""; ""Random Sums of Independent Random Variables with Non-random Centering""; ""Functions of Non-linear Statistics""; ""Uniform and Non-uniform Randomized Concentration Inequalities""
,
""Appendix""
Additional Edition:
ISBN 9783642150067
Additional Edition:
Buchausg. u.d.T. Chen, Louis H. Y., 1940 - Normal approximation by Stein's method Heidelberg : Springer, 2011 ISBN 9783642150067
Additional Edition:
ISBN 9783642265655
Additional Edition:
ISBN 3642265650
Language:
English
Subjects:
Mathematics
Keywords:
Wahrscheinlichkeitsverteilung
;
Approximation
;
Stein-Schätzung
DOI:
10.1007/978-3-642-15007-4
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
Chen, Louis H. Y. 1940-
